What is Integrals: Definition and 1000 Discussions

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

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  1. S

    Difference Between Surface Integrals and Surface area using double integrals .

    Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...
  2. C

    Imaginary components of real integrals

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  3. F

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  4. R

    Comparison test on second species integrals

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  5. T

    On Taylor Series Expansion and Complex Integrals

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  6. B

    Source for calculating Incomplete Elliptical Integrals

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  7. P

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  8. B

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  9. M

    Evalute the following integrals

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  10. C

    Can Gaussian integrals be done with half integrals?

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  11. J

    Given 2 Integrals, How to solve other Integrals?

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  12. Kushwoho44

    Order of Indefinite Double Integrals

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  13. R

    Integrals and exponential growth problem

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  14. Vorde

    Work-Energy Theorem with Line Integrals

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  15. A

    A question about Upper Darboux Integrals

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  16. C

    Vector Calculus Question about Surface Integrals

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  17. M

    Conceptual question on greens theorem/line integrals

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  18. A

    What is the geometrical significance of definite integrals of vector functions?

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  19. D

    Integrals look easy but I'm still confused

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  20. A

    What Are Riemann-Stieltjes Integrals and How Can We Visualize Them?

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  21. A

    Change of Variable in multiple Integrals

    Homework Statement Let D be the triangular region in the xy-plane with the vertices (1, 2), (3, 6), and (7, 4). Consider the transformation T : x = 3u − 2v, y = u + v. (a) Find the vertices of the triangle in the uv-plane whose image under the transformation T is the triangle D. (b)...
  22. G

    Integrals going to 0 implies functions go to 0?

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  23. D

    General solution of double integrals

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  24. M

    Integrals over a transformed region

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  25. alane1994

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  26. O

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  27. U

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  28. N

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  29. M

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  30. L

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  31. F

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  32. M

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  33. R

    Integrals of Exponential(Polynomial(x)) dx Form

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  34. E

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  35. R

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  36. S

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  37. C

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  38. S

    Laurent Series-Finding Contour Integrals

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  39. M

    Evaluate integrals using modified Bessel function of the second kind

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  40. R

    Finding Whether Improper Integrals Converge

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  41. G

    Using complex numbers for evaluating integrals

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  42. J

    MHB What is the method for integrating 1/(1+x^n) using roots of polynomials?

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  43. C

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  44. R

    Numerical Integration of Double integrals

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  45. L

    Phase Space, Velocity, Integrals

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  46. majormaaz

    Acceleration of a brick, involving integrals

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  47. D

    Finding the amount of work done (line integrals)

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  48. Kawakaze

    Help with revision - area integrals

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  49. D

    Fundamental theorem for line integrals

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  50. phosgene

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